Develop an efficient solver for the parameter polynomial system arising in Riccati validation

Develop an efficient algorithm to solve, in general and in practice, the polynomial system in the parameters g obtained by substituting a parametrized rational candidate u(x)=P(x)/Q(x) into the Riccati Mahler equation ℓ_r(x) u(x) u(x^b) ⋯ u(x^{b^{r−1}})+⋯+ℓ_1(x) u(x)+ℓ_0(x)=0 and equating coefficients. The system has degree r in the parameters g and arises in the Hermite–Padé validation step for certifying which parameter specializations yield true rational solutions without increasing the truncation order σ.

Background

In their Hermite–Padé-based algorithm, the authors generate parametrized rational candidates u=P/Q for solutions of the Riccati Mahler equation. To certify candidates without increasing the series truncation order σ, one natural idea is to substitute P/Q into the Riccati equation and identify coefficients to zero, which yields a polynomial system in the parameters g that determine P and Q.

Although this approach would separate correct parameter specializations from incorrect ones at fixed σ, the resulting system has degree r in g and the authors do not know how to solve it efficiently. An efficient solver would accelerate validation and potentially improve the practical performance of the algorithm.